Structure-Preserving Flows of Symplectic Matrix Pairs

STRUCTURE-PRESERVING FLOWS OF SYMPLECTIC MATRIX PAIRS

YUEH-CHENG KUO∗, WEN-WEI LIN† , AND SHIH-FENG SHIEH‡

Abstract. We construct a nonlinear differential equation of matrix pairs (M(t), L(t)) that are invariant (Structure-Preserving Property) in the class of symplectic matrix pairs X12 0 IX11 (M, L) = S2, S1 X = [Xij ]1≤i,j≤2 is Hermitian , X22 I 0 X21 where S1 and S2 are two fixed symplectic matrices. Furthermore, its solution also preserves deflating subspaces on the whole orbit (Eigenvector-Preserving Property). Such a flow is called a structure-preserving flow and is governed by a Riccati differential equation (RDE) of the form > > W˙ (t) = [−W (t),I]H [I,W (t) ] , W (0) = W0, for some suitable Hamiltonian matrix H . We then utilize the Grassmann manifolds to extend the domain of the structure-preserving flow to the whole R except some isolated points. On the other hand, the Structure-Preserving Doubling Algorithm (SDA) is an efficient numerical method for solving algebraic Riccati equations and nonlinear matrix equations. In conjunction with the structure-preserving flow, we consider two special classes of symplectic pairs: S1 = S2 = I2n and S1 = J , S2 = −I2n as well as the associated algorithms SDA-1 k−1 and SDA-2. It is shown that at t = 2 , k ∈ Z this flow passes through the iterates generated by SDA-1 and SDA-2, respectively. Therefore, the SDA and its corresponding structure-preserving flow have identical asymptotic behaviors. Taking advantage of the special structure and properties of the Hamiltonian matrix, we apply a symplectically similarity transformation to reduce H to a Hamiltonian Jordan canonical form J. The asymptotic analysis of the structure-preserving flows and RDEs is studied by using eJt. Some asymptotic dynamics of the SDA are investigated, including the linear and quadratic convergence.

1. Introduction. The following three important types of algebraic Riccati equations appear in many diﬀerent ﬁelds of applied mathematics, sciences and engineering. • Continuous-time Algebraic Riccati Equation (CARE) [24, 28]:

−XGX + AH X + XA + H = 0; (1.1)

• Discrete-time Algebraic Riccati Equation (DARE) [24, 28]:

X = AH X(I + GX)−1A + H; (1.2)

• Nonlinear Matrix Equation (NME) [11]:

X + AH X−1A = Q, (1.3) where A, H, G, Q ∈ Cn×n with G = GH , H = HH and Q = QH . The CARE (1.1) and DARE (1.2) with G and H being positive semi-definite arise in the linear quadratic (LQ) optimal control problems in continuous- and discrete-time, respectively (see [24, 28]). In the H∞-control problems, G and H in (1.1) or (1.2) can possibly be indefinite [13, 28]. In addition, solutions of a CARE are the equilibria of Riccati differential equations (RDEs) that arise frequently from optimal controls [23, 25], or two-point boundary value problems [9, 10]. The NME (1.3) occurs in applications such as analysis of ladder networks, dynamic programming and stochastic filtering

∗Department of Applied Mathematics, National University of Kaohsiung, Kaohsiung, 811, Taiwan ([email protected]) †Department of Applied Mathematics, National Chiao Tung University, Hsinchu 300, Taiwan ([email protected]) ‡Department of Mathematics, National Taiwan Normal University, Taipei 116, Taiwan ([email protected]) 1 (see [2, 11, 12] and the references therein). Nonclassical Riccati-type equations for which the conjugate transpose in (1.3) is replaced by the complex transpose also have various applications; for example, in the vibration analysis of fast trains [16] and the Green’s function calculation in nano research [15]. The Structure-Preserving Doubling Algorithms (SDAs) [3, 20, 27] are employed for solving the stabilizing solutions of the three Riccati-type equations (1.1)-(1.3). We now describe these SDAs for solving DARE/CARE and NME. Ak 0 IGk • For solving DAREs (1.2), the pairs (Mk, Lk) = , H −Hk I 0 Ak are generated by Algorithm SDA-1 [20, 27]. Let A1 = A, G1 = G, and H1 = H. For k = 1, 2,..., compute

−1 Ak+1 = Ak(I + GkHk) Ak, −1 H Gk+1 = Gk + AkGk(I + HkGk) Ak , H −1 Hk+1 = Hk + Ak (I + HkGk) HkAk. • For solving CAREs (1.1), one can transform it into a DARE (1.2) by using the Cayley transform and a left equivalence transform [4]. Then Algorithm SDA-1 can be employed to ﬁnd the desired solutions of CAREs. Ak 0 −Pk I • For solving NMEs, the pairs (Mk, Lk) = , H are Qk −I Ak 0 generated by Algorithm SDA-2 [3, 27]. Let A1 = A, Q1 = Q, and P1 = 0. For k = 1, 2,..., compute

−1 Ak+1 = Ak(Qk − Pk) Ak, H −1 Qk+1 = Qk − Ak (Qk − Pk) Ak, −1 H Pk+1 = Pk + Ak(Qk − Pk) Ak .

∞ Assume that the sequences {(Mk, Lk)}k=1 generated by the SDAs are well- ∞ ∞ defined. The sequences {(Hk,Gk,Ak)}k=1 by SDA-1 and {(Qk,Pk,Ak)}k=1 by SDA-2 ∗ ∗ ∗ ∗ converge quadratically to (H ,G , 0) and (Q ,P , 0), respectively, if (M1, L1) has no unimodular eigenvalues [3, 27]. The sequences by SDA-1 and SDA-2 converge linearly of the rate 1/2 if the partial multiplicities of the unimodular eigenvalues of (M1, L1) are all even [3, 20]. Here H∗ and G∗ are the stabilizing solutions of (1.2) and its dual equation: Y = AY (I + HY )−1AH + G, respectively, with the spectral radius ρ((I + GH∗)−1A) < 1. Q∗ and P ∗ are the maximal and the minimal solutions of (1.3), respectively, with Q∗ − X and X − P ∗ being positive semi-definite, where X is any solution of (1.3). Furthermore, the SDA has two preserving properties [22, 27]: Eigenvector-Preserving Property: For each case above, if MkU = LkUS or 2n×r r×r MkVT = LkV , where U, V ∈ C are of full column rank and S, T ∈ C , 2 2 then Mk+1U = Lk+1US or Mk+1VT = Lk+1V , i.e., the SDA preserves the deflating subspaces for each k and squares the eigenvalue matrix; ∞ Structure-Preserving Property: The sequences of pairs {(Mk, Lk)}k=1 generated by SDA-1 and SDA-2 are symplectic (see Definition 2.1) and are, respectively, contained in the sets

A 0 IG = , |A, H = HH ,G = GH ∈ n×n , (1.4a) S1 HI 0 AH C

2 and A 0 PI = , |A, Q = QH ,P = P H ∈ n×n . (1.4b) S2 Q −I AH 0 C

To study the symplectic pairs, we first quote the following useful theorem. Theorem 1.1 (see [29]). Let (M, L) be a regular symplectic pair with M, L ∈ 2n×2n C . Then there exist symplectic matrices S1, S2 and a Hermitian matrix X = X11 X12 l.e. X12 0 IX11 such that (M, L) ∼ S2, S1 . Here we say X21 X22 X22 I 0 X21 l.e. (A1,B1) ∼ (A2,B2), if A1 = CA2, B1 = CB2 for some invertible matrix C. Theorem 1.1 provides us a classification for symplectic pairs. Specifically, let S1, S2 be symplectic and H(2n) be the set of all 2n × 2n Hermitian matrices. We denote the class of symplectic pairs generated by S1, S2 as X12 0 IX11 X11 X12 SS1,S2 = S2, S1 | ∈ H(2n) . (1.5a) X22 I 0 X21 X21 X22

It is easily seen that each pair (M, L) ∈ SS1,S2 is symplectic. We deﬁne a bijective

transformation TS1,S2 : H(2n) → SS1,S2 by X12 0 IX11 TS1,S2 (X) = S2, S1 . (1.5b) X22 I 0 X21

Therefore, the invariant sets S1 and S2 for SDA-1 and SDA-2, respectively, can be

rewritten as S1 = SI2n,I2n and S2 = SJ ,−I2n . Note that S1 *S2, S2 *S1.

The purpose of this paper is to focus on the flows (M(t), L(t)) on a specified SS1,S2 (i.e., the Structure-Preserving Property) that have Eigenvector-Preserving Property. Such a flow is called a structure-preserving flow. For two special classes S1 and S2 of symplectic pairs, the structure-preserving flow passes through a sequence of iterates generated by SDAs. Finding a smooth curve with a specific structure that passes through a sequence of iterates generated by some numerical algorithm is a popular topic studied by many researchers, especially in the study of the so- called Toda flow that links matrices/matrix pairs generated by QR/QZ-algorithm [5, 6, 7, 32]. In [22], a parameterized curve is constructed in S2 passing through the iterates generated by the fixed-point iteration, the SDA and the Newton’s method with some additional conditions. Our first main contribution in this paper is to construct a nonlinear differential equation associated with the structure-preserving flow satisfying (M(1), L(1)) =

(M1, L1) ∈ SS1,S2 . Before the description of this main result, we first quote the preliminary result in Theorem 2.7. Suppose that ind∞(M1, L1) 6 1. There exist an 2n×2ˆn 2n×` 2n×` invertible matrix U = [U1|U0,U∞] ∈ C × C × C and a Hamiltonian 2ˆn×2ˆn Hb Hb ∈ C such that M1U0 = 0, L1U∞ = 0, and M1U1 = L1U1e . Here, the spaces spanned by U0, U∞, and U1 represent the eigenspaces corresponding to zero eigenvalues, infinity eigenvalues, and finite nonzero eigenvalues of the pair, respectively.

Main Result I (Theorem 3.3). Let (M1, L1) ∈ SS1,S2 be a regular symplectic pair with ind (M , L ) 1 and X = T −1 (M , L ). There exists a Hamiltonian ∞ 1 1 6 1 S1,S2 1 1 H ∈ C2n×2n such that the solution of the initial value problem (IVP):

H X˙ (t) = M(t)HJM(t) ,X(1) = X1, (1.6)

3 Hbt for t ∈ (t0, t1), satisﬁes M(t)U0 = 0, L(t)U∞ = 0 and M(t)U1 = L(t)U1e , where

(M(t), L(t)) = TS1,S2 (X(t)). This implies that the pair (M(t), L(t)) preserves its deflating subspaces as t varies, i.e., the flow satisfies the so-called Eigenvector- Preserving Property. We further show that the IVP (1.6) is governed by an RDE. We then adopt the Grassmann manifold and Radon’s lemma ([31] or see Theorem 3.14) to extend the domain of the structure-preserving flow to the whole R except some isolated points. For the S2 case, the phase portrait of the extended flow is the parameterized curve constructed in [22]. Our second main contribution shows that the extended flow passes through the k−1 k-th iterate generated by the SDA with initial pair (M1, L1) at t = 2 . Main Result II (Theorem 3.19). Let (M1, L1) ∈ S1 or S2 defined in (1.4) be ∞ regular with ind∞(M1, L1) 6 1. Suppose {(Mk, Lk)}k=1 is the sequence generated by k−1 k−1 the SDAs. Then (Mk, Lk) = (M(2 ), L(2 )), where (M(t), L(t)) = TS1,S2 (X(t)) and X(t) is the extended solution of the IVP (1.6). Therefore, the SDA and its associated structure-preserving flow have identical asymptotic behaviors, including the stability, instability, and periodicity of the dynamics. By applying the asymptotic analysis of the flow to the Main Result II, our third main contribution concerns the convergence of the SDAs. Main Result III (Theorems 4.2, 4.4 and 4.7). Let (M1, L1) ∈ S1 or S2 be 2n×2n regular with ind∞(M1, L1) 6 1 and H ∈ C be given in Main Result I. Let {(M , L )}∞ be generated by the SDAs and [Xk ] = T −1 (M , L ). Then k k k=1 ij 16i,j62 S1,S2 k k k (i) each Xij converges quadratically if H has no pure imaginary eigenvalues; k (ii) each Xij converges linearly if each pure imaginary eigenvalue of H has only even partial multiplicities; (iii) suppose that H ∈ C2n×2n has only one eigenvalue iα with two odd partial k k k k multiplicities. Then X11, X22 converge linearly and X12, X21 approach closed curves in Cn×n. In the latter case, the closed curves consist of rank one matrices. Similar convergence analysis to assertions Main Result III (i) and (ii) have been carried out in [3, 16] and [20], respectively. In both cases, the proof of convergence for the matrix H is only applicable when the Jordan blocks, with pure imaginary eigenvalues, are of even sizes. We hereby present an asymptotic behavior of SDA in Main Result III (iii) for these Jordan blocks are of odd sizes. This paper is organized as follows. In Section 2, we introduce some preliminary results of the eigenstructures of regular symplectic pairs. In Subsection 3.1, we prove Main Result I (Theorem 3.3). The connection between the structure-preserving flow and RDE is studied in Subsection 3.2. In Subsection 3.3, we then apply the Grassmann manifold to extend the flow to the whole R. In Subsection 3.4, we prove Main Result II (Theorem 3.19). In Section 4, we give a brief description for the proof of Main Result III (Theorems 4.2, 4.4 and 4.7). Some numerical experiments are presented in Section 5. 2. Eigenstructures of regular symplectic pairs. We first introduce some notation, definitions and the algebraic structures that we consider in this paper. Let 0 In Jn = , where In is the n × n identity matrix. For convenience, we use J −In 0 for Jn by dropping the subscript “n” if the order of Jn is clear in the context. Two subspaces U and V of C2n are called J -orthogonal if uH J v = 0 for each u ∈ U and v ∈ V. A subspace U of C2n is called isotropic if xH J y = 0 for any x, y ∈ U. An 4 n-dimensional isotropic subspace is called a Lagrangian subspace. The jth column of an identity matrix I is denoted by ej and k · k denotes the Forbenious norm. Definition 2.1. 1. A matrix H ∈ C2n×2n is Hamiltonian if HJ = (HJ )H . 2. A matrix S ∈ C2n×2n is symplectic if SJSH = J . 3. A matrix pair (M, L) ∈ C2n×2n ×C2n×2n is symplectic if MJ MH = LJ LH . Denote by Sp(n) the multiplicative group of all 2n × 2n symplectic matrices and by H(2n) the set of all 2n × 2n Hermitian matrices. A matrix pair (A, B) with A, B ∈ Cn×n is said to be regular if det(A − λB) 6= 0 for some λ ∈ C. Note that the matrix pair (A, B) is said to have eigenvalues at infinity if B is singular. To see this, write the eigenvalue problem Ax = λBx in the reciprocal form Bx = λ−1Ax. In the case B is singular, we have Bx = 0Ax whenever x is a null vector of B. This means that x is an eigenvector of the eigenvalue problem corresponding to eigenvalue −1 n×n n×n λ = 0, i.e., λ = ∞. The matrix pairs (A1,B1) and (A2,B2) ∈ C ×C are said l.e. to be left equivalent, denoted by (A1,B1) ∼ (A2,B2), if A1 = CA2, B1 = CB2 for some invertible matrix C. It is well-known that for a regular matrix pair (A, B) there are invertible matrices P and Q which transform (A, B) to the Kronecker canonical form [14] as

J 0 I 0 P AQ = ,PBQ = , (2.1) 0 I 0 N where J is a Jordan matrix corresponding to the finite eigenvalues of (A, B) and N is a nilpotent Jordan matrix corresponding to the infinity eigenvalues. The index of a matrix pair (A, B) is the index of nilpotency of N, i.e., the matrix pair (A, B) is of ν−1 ν index ν, denoted by ν = ind∞(A, B), if N 6= 0 and N = 0. Remark 2.1. Let (A, B) be a regular pair. Then ind∞(A, B) = 0 if B is invertible and ind∞(A, B) = 1 if the infinity eigenvalues of (A, B) are semi-simple. 2.1. Eigenvector-Preserving Property. Motivated by the Eigenvector-Preserving Property of iterations of SDAs, we extend this property to a flow. Let (M1, L1) ∈

SS1,S2 be regular and ind∞(M1, L1) 6 1. A flow {(M(t), L(t)) | t ∈ R} ⊆ SS1,S2 with (M(1), L(1)) = (M1, L1) having the Eigenvector-Preserving Property can be stated as follows: Assume that M1U0 = 0, L1U∞ = 0, and M1U1 = L1U1S, where U = 2n×2n [U1,U0,U∞] ∈ C and S are invertible. Then t M(t)U0 = 0, L(t)U∞ = 0, M(t)U1 = L(t)U1S (2.2) hold. We shall note that the flow (M(t), L(t)) in (2.2) preserves eigenvectors whenever S in (2.2) is diagonalizable, otherwise, it preserves the deflating subspaces only. Here in (2.2), St, for t ∈ R, is defined in the sense of a matrix function. Because S invertible, it follows from [17, Definition 1.11 and Theorem 1.17] that the matrix function St is well-defined. We shall show in Theorem 2.2 that the invertibility of the matrix S can result from the condition ind∞(M1, L1) 6 1. In this case, (2.2) can have an alternative form as (2.1) in which P M(t)Q = J t ⊕ I and P L(t)Q = I ⊕ 0.

2.2. Regular symplectic pairs with ind∞(M, L) 6 1. The proof of the following theorem can be found in [21, 26] Theorem 2.2. Suppose (M, L) is a regular symplectic pair with M, L ∈ C2n×2n and ind∞(M, L) 6 1. Then there is nˆ 6 n such that rank(M) = rank(L) = n +n ˆ. In 5 2n×2ˆn 2n×` 2n×` addition, there exist an invertible matrix U = [U1|U0,U∞] ∈ C × C × C with ` = n − nˆ and a symplectic matrix Sb ∈ C2ˆn×2ˆn such that

H U JnU = Jnˆ ⊕ J` (2.3) and

MU0 = 0, LU∞ = 0, MU1 = LU1Sb. (2.4)

Remark 2.3. Theorem 2.2 shows that the assumption of the Eigenvector Pre- serving Property holds if ind∞(M1, L1) 6 1. Note that the matrix Sb in Theorem 2.2 is symplectic. There is a Hamiltonian matrix Hb satisfying eHb = Sb (see e.g., [19, 21, 30]). Using Hb, we shall construct a Hamiltonian matrix H which has invariant subspaces spanned by U0, U∞, and U1. Theorem 2.4. Suppose (M, L) is a regular symplectic pair with M, L ∈ C2n×2n and ind∞(M, L) 6 1. Let the matrices U and Sb be given as in Theorem 2.2, and Hb ∈ C2ˆn×2ˆn be the Hamiltonian matrix such that eHb = Sb. Then the matrix H 0 H = U b (J ⊕ J )H UH J (2.5) 0 0 nˆ `

is Hamiltonian. H H H H H Proof. Since Hb is Hamiltonian, we have HJ = −U(HJb nˆ ⊕0)U = J J U(Jnˆ Hb ⊕ 0)UH = J H HH . Hence, H is Hamiltonian. Remark 2.5. Suppose that (M, L) is a real regular symplectic pair. Then U and Sb can be chosen real. In [8], under the assumptions (i) Sb has an even number of Jordan blocks of each size relative to every negative eigenvalue; (ii) the size of two identical Jordan blocks corresponding to eigenvalue −1 is odd; it is shown that there is a real Hamiltonian matrix Hb such that eHb = Sb. Hence, under these assumptions, the Hamiltonian H deﬁned in (2.5) can be chosen real. Suppose that L is invertible. It follows from Theorem 2.2 that M is also invertible. −1 Therefore, U0 and U∞ in (2.4) are absent. On the other hand, the matrix L M is symplectic, hence there exists a Hamiltonian matrix H such that eH = L−1M, i.e., M = LeH. For the case that L is singular and (M, L) is a regular symplectic pair with ind∞(M, L) = 1, it is natural to ask whether there is a Hamiltonian matrix H such that M = LeH. However, this is not true (see Lemma 2.6 for the necessary H condition). The pair (M, L) satisﬁes the equality MΠ0 = LΠ∞e for some suitable matrices Π0 and Π∞. To see this, we need the following lemma. Lemma 2.6. Suppose that (M, L) is a regular symplectic pair. If M = LW for some nonsingular W , then both M and L are invertible. Proof. From Theorem 1.1, there are matrices S1, S2 ∈ Sp(n) and X = [Xij]1≤i,j≤2 ∈ X12 0 IX11 H(2n) such that M = C S2, L = C S1, where C is nonsin- X22 I 0 X21 X12 0 IX11 gular. Suppose that M = LW . Then we have S2 = S1W . X22 I 0 X21 Since S1, S2 and W are nonsingular, it is easily seen that X12 and X21 are nonsingular. Thus, M and L are invertible. 6 Theorem 2.7. Suppose (M, L) is a regular symplectic pair with M, L ∈ C2n×2n and ind∞(M, L) 6 1. Let the matrices U and H be given as in Theorems 2.2 and 2.4, respectively. Let

−1 −1 Π0 = U(I2ˆn ⊕ I` ⊕ 0)U and Π∞ = U(I2ˆn ⊕ 0 ⊕ I`)U , (2.6)

−1 H H H where U = (Jnˆ ⊕ J`) U J . Then we have MΠ0 = LΠ∞e . 2 Remark 2.8. Note that both Π0 and Π∞ are idempotent, i.e., Π0 = Π0 and 2 Π∞ = Π∞. Here Π0 (Π∞, respectively) is a projection onto the space spanned by the eigenspace corresponding to the finite eigenvalues (nonzero eigenvalues, respectively) along the space spanned by U∞ (by U0, respectively). In addition, if ind∞(M, L) = 0, H i.e., M and L are invertible and nˆ = n, then Π0 = Π∞ = I. Therefore, M = Le with some appropriate Hamiltonian matrix H. This coincides with Lemma 2.6. Proof. [Proof of Theorem 2.7] From (2.4) and (2.6), we have −1 −1 −1 MΠ0 = [LU1S|b 0, 0]U = L[U1|U0,U∞](Sb ⊕ 0 ⊕ I`)U = LΠ∞U(Sb ⊕ I2`)U . Hb H Hb 0 −1 −1 It follows from (2.5) and the fact e = Sb that e = U(e ⊕e )U = U(Sb ⊕I2`)U . The assertion follows. To make the correspondence between the constructed matrices in the previous lemmas/theorems and the symplectic pairs (M, L), we use the following notations throughout this paper. Definition 2.2. For a given regular symplectic pair (M, L) with ind∞(M, L) 6 1. Let nˆ := rank(M) − n and ` = n − nˆ. We define U(M, L) := [U1|U0,U∞] ∈ C2n×2ˆn × C2n×` × C2n×` and Sb(M, L) ∈ C2ˆn×2ˆn that satisfy (2.3) and (2.4). Let Hb(M, L) be the matrix that satisfies eHb(M,L) = Sb(M, L), and let H(M, L) and Π0(M, L), Π∞(M, L) be the matrices defined in (2.5) and (2.6), respectively. 2.3. A perturbation result for symplectic pairs. We now provide a perturbation theory for a regular symplectic pair (M, L) with ind∞(M, L) = 1. In this case, the infinity eigenvalues of the pair (M, L) exist and are semi-simple, and hence, so does the zero eigenvalues, by Theorem 2.2. After a small perturbation of order O(ε) in a specific direction, the perturbed pair (Mε, Lε) preserves the deflating subspaces ε ε spanned by U0, U∞, and U1. Moreover, finite and nonzero eigenvalues of (M , L ) are remained the same as that of (M, L). Zero and infinity eigenvalues of (M, L) are perturbed to finite eigenvalues of (Mε, Lε) of order O(ε) and of order O(1/ε), respectively. Theorem 2.9. Suppose (M, L) is a regular symplectic pair with M, L ∈ C2n×2n and ind∞(M, L) = 1. Let U = U(M, L) and Sb = Sb(M, L) be given as in Defini- ε `×` ε tion 2.2 and let Φ ∈ C be a family of nonsingular matrices with kΦ k 6 ε for each ε > 0. If

Mε = M + ∆Mε, Lε = L + ∆Lε, (2.7)

ε εH H ε ε H ε ε where ∆M = −LU0Φ U∞J , ∆L = MU∞Φ U0 J , then (M , L ) is a regular symplectic pair with Lε being invertible. Moreover, Mε and Lε satisfy

ε ε εH ε ε ε ε ε M U0 = L U0Φ , M U∞Φ = L U∞, M U1 = L U1Sb, (2.8) and (Mε, Lε) → (M, L) as ε → 0. Proof. From (2.3), it holds that

H H H H U0 J U∞ = I,U∞J U0 = −I,U∞J U∞ = U0 J U0 = 0. (2.9) 7 For each ε > 0, from (2.7) and (2.9) it holds that MεJMεH = (M + ∆Mε)J (M + ∆Mε)H = MJ MH + MJ ∆MεH + ∆MεJMH + ∆MεJ ∆MεH H ε H H εH H H = LJ L − MU∞Φ U0 L + LU0Φ U∞M = LJ LH + ∆LεJLH + LJ ∆LεH + ∆LεJLH + ∆LεJ ∆LεH = (L + ∆Lε)J (L + ∆Lε)H = LεJLεH . That is, (Mε, Lε) forms a symplectic pair. Now, we show that Lε is invertible. Since (M, L) is a regular symplectic pair, there exists a nonzero constant λ0 such that M − λ0L is invertible. Using the fact that U is nonsingular, it follows from (2.4) that h i (M − λ0L)U = [MU1 − λ0LU1, −λ0LU0, MU∞] = LU1(Sb − λ0I), −λ0LU0, MU∞

ε is nonsingular, and hence, Sb − λ0I is also invertible. Since Φ is nonsingular, from H (2.7) and (2.9) together with the fact that U0 J U1 = 0, we have

ε ε −1 −1 ε L U = [LU1, LU0, MU∞Φ ] = (M − λ0L)U (Sb − λ0I) ⊕ (−λ0) I ⊕ Φ

is invertible and then Lε is invertible. Hence, (Mε, Lε) is a regular symplectic pair. From (2.4) and (2.9), we have

ε ε εH H εH ε εH M U0 = (M + ∆M )U0 = −LU0Φ U∞J U0 = LU0Φ = L U0Φ , ε ε ε H ε ε ε L U∞ = (L + ∆L )U∞ = MU∞Φ U0 J U∞ = MU∞Φ = M U∞Φ , ε ε ε ε M U1 = (M + ∆M )U1 = MU1 = LU1Sb = (L + ∆L )U1Sb = L U1Sb. ε ε ε Thus, equations of (2.8) hold. Since kΦ k 6 ε,(M , L ) → (M, L) as ε → 0.

Corollary 2.10. Suppose (M, L) ∈ SS1,S2 is a regular symplectic pair with ε ε ind∞(M, L) 6 1. Let Φ be nonsingular with kΦ k 6 ε for each ε > 0 sufficiently ε ε ε ε small, and M , L be given as in Theorem 2.9. Then there exists (Mf , Le ) ∈ SS1,S2 ε ε l.e. ε ε for ε > 0 sufficiently small, such that (M , L ) ∼ (Mf , Le ). Moreover, for each ε > 0 sufficiently small, Mfε and Leε are invertible satisfying (2.8) and (Mfε, Leε) → (M, L) as ε → 0. X12 0 IX11 Proof. Since (M, L) ∈ SS1,S2 , it holds that M = S2, L = S1, X22 I 0 X21 ε where [Xij]1≤i,j≤2 is Hermitian. Since kΦ k 6 ε for ε > 0 sufficiently small, from (2.7) we have ε X12 + O(ε) O(ε) ε I + O(ε) X11 + O(ε) M = S2, L = S1. X22 + O(ε) I + O(ε) O(ε) X21 + O(ε) Applying row operations to (Mε, Lε) we have " # " # ! ε ε l.e. Xe12(ε) 0 I Xe11(ε) ε ε (M , L ) ∼ S2, S1 ≡ Mf , Le , Xe22(ε) I 0 Xe21(ε)

ε ε where Xeij(ε) = Xij + O(ε) for 1 6 i, j 6 2. Hence, (Mf , Le ) → (M, L) as ε → 0. l.e. Using the fact that (Mε, Lε) ∼ (Mfε, Leε), it follows from Theorem 2.9 that Mfε and Leε are invertible, and satisfy the equalities of (2.8). Since (Mfε, Leε) is symplectic and ε ε [Xeij(ε)]1≤ij≤2 is Hermitian, we have (Mf , Le ) ∈ SS1,S2 for ε > 0 suﬃciently small. 8 3. Structure-preserving ﬂows.

3.1. Construction of structure-preserving ﬂows. Suppose that (M1, L1) is a regular symplectic pair with ind∞(M1, L1) 6 1. From Theorem 1.1, there exist two

symplectic matrices S1 and S2 such that (M1, L1) ∈ SS1,S2 . In this subsection we shall construct a differential equation with (M1, L1) as an initial matrix pair whose solution is the structure-preserving flow. We first prove the case for which L1 is invertible. 2n×2n Theorem 3.1. Let S1, S2 ∈ Sp(n), H ∈ C be Hamiltonian and X1 ∈ H(2n). Suppose X(t), for t ∈ (t0, t1) and t0 < 1 < t1, is the solution of the IVP: H X˙ (t) = M(t)HJM(t) ,X(1) = X1, (3.1)

where (M(t), L(t)) = TS1,S2 (X(t)). If the pair (M1, L1) ≡ (M(1), L(1)) satisﬁes H1 2n×2n M1 = L1e for some Hamiltonian H1 ∈ C , then

M(t) = L(t)eH1 eH(t−1) (3.2)

for all t ∈ (t0, t1). 1 Proof. Partition X1 = [Xij]1≤i,j≤2 and X(t) = [Xij(t)]1≤i,j≤2. From the assump- H1 tion M1 = L1e and Lemma 2.6, we see that both M(1) and L(1) are invertible. On the other hand, the solution X(t) of the IVP (3.1) is continuous. Therefore, we denote (t˜0, t˜1) the connected component of the open set {t ∈ (t0, t1)|det(M(t)) 6= 0, det(L(t)) 6= 0} that contains t = 1. We ﬁrst show that assertion (3.2) holds for t ∈ (t˜0, t˜1). By the fact that X12(t) 0 IX11(t) M(t) = S2, L(t) = S1, X22(t) I 0 X21(t) we have   0 In ˙ ˙ H H H ˙ X12 0 0 X11  −X12 −X22  ˙ ˙ JM X = ˙ ˙  H H  = [M, L] H . (3.3) X22 0 0 X21  −X11 −X21  −J L In 0

Plugging (3.3) into the ﬁrst equation of (3.1) and multiplying M−H J H from the right to the resulting equation, we have

I [M˙ , L˙] = MH, t ∈ (t˜ , t˜ ). (3.4) JLH M−H J 0 1

Since MJ MH = LJ LH and both M and L are invertible, (3.4) becomes M˙ − L˙(L−1M) = MH. Multiplying L−1 from the left of the last equation, we thus obtain −1 ˙ −1 ˙ −1 −1 d −1 −1 L M − (L LL )M = L MH. This coincides with dt (L M) = (L M)H. H1 Therefore, using the initial condition in (3.1) and the fact M1 = L1e , it follows −1 H1 H(t−1) that L(t) M(t) = e e for t ∈ (t˜0, t˜1). Hence, assertion (3.2) holds. Now we claim that t˜0 = t0 and t˜1 = t1. We only prove the case t˜1 = t1. Suppose that t˜1 < t1. Then M(t˜1) and L(t˜1) exist. Since (t˜0, t˜1) is a connected component of {t ∈ (t0, t1)|det(M(t)) 6= 0, det(L(t)) 6= 0}, M(t˜1) and L(t˜1) are singular. Using (3.2) ˜ ˜ ˜− ˜ ˜ H1 H(t1−1) H1 H(t1−1) and taking the limit t → t1 , we have M(t1) = L(t1)e e . Since e e is invertible, M(t˜1) and L(t˜1) are invertible by Lemma 2.6. This is a contradiction. Hence, t˜0 = t0 and t˜1 = t1. 9 Remark 3.2. (i) In Theorem 3.1, since X1 and HJ are Hermitian, it is easily seen that X(t) is also Hermitian for t ∈ (t0, t1). From the deﬁnition that

(M(t), L(t)) = TS1,S2 (X(t)), we have that the curve {(M(t), L(t))|t ∈ (t0, t1)} ⊂

SS1,S2 . (ii) Suppose (M1, L1) is a real symplectic pair. Under the conditions in Re- mark 2.5, H1 can be chosen real. If H is also real, then the curve {(M(t), L(t))|t ∈

(t0, t1)} ⊂ SS1,S2 is real. The following theorem is the detail version of Main Result I. Theorem 3.3. Let S1, S2 ∈ Sp(n) and X1 ∈ H(2n) be given such that the

symplectic pair (M1, L1) = TS1,S2 (X1) is regular with ind∞(M1, L1) 6 1. Let the idempotent matrices Π0 = Π0(M1, L1), Π∞ = Π∞(M1, L1) and the Hamiltonian matrix H = H(M1, L1) be deﬁned in Deﬁnition 2.2 such that (from Theorem 2.7)

H M1Π0 = L1Π∞e . (3.5)

If X(t), for t ∈ (t0, t1), t0 < 1 < t1, is the solution of the IVP (3.1), then

Ht M(t)Π0 = L(t)Π∞e , (3.6) or equivalently,

Hbt M(t)U0 = 0, L(t)U∞ = 0, M(t)U1 = L(t)U1e , (3.7)

where (M(t), L(t)) = TS1,S2 (X(t)), for all t ∈ (t0, t1). Remark 3.4. Note that (i) if M1 and L1 are invertible, this implies Π0 = Π∞ = I, and hence the result of Theorem 3.3 is consistent with Theorem 3.1 in which H1 is replaced by H; and (ii) from Definition 2.2 of H, Π0 and Π∞, it is easily seen that the invariant properties (3.6) and (3.7) are equivalent. This shows that the flow (M(t), L(t)) satisfies Eigenvector-Preserving Property. Actually, this flow (M(t), L(t)) is the structure-preserving flow with the initial (M1, L1). Proof. [Proof of Theorem 3.3] Applying Theorem 2.9 and Corollary 2.10 with ε ε ε Φ = εI, we see that (M1 + ∆M , L1 + ∆L ) is left equivalent to the symplectic pair

1ε 1ε ε ε ε ε X12 0 IX11 (M1, L1) ≡ (Mf , Le ) = 1ε S2, 1ε S1 ∈ SS1,S2 X22 I 0 X21

ε ε ε ε for ε suﬃciently small. In addition, M1 and L1 are invertible for ε > 0 and (M1, L1) → ε ε ε X11(t) X12(t) (M1, L1) as ε → 0. Let X (t) = ε ε be the solution of the IVP X21(t) X22(t) ˙ ε ε ε H ε ε X (t) = M (t)HJM (t) ,X (1) = X1 ,

1ε 1ε ε X11 X12 ε ε ε where X1 = 1ε 1ε and (M (t), L (t)) = TS1,S2 (X (t)). By the continuous X21 X22 dependence of the solution on the initial condition of the IVP (see e.g. Section 8.4 in [18]), we have (Mε(t), Lε(t)) → (M(t), L(t)) as ε → 0. On the other hand, it follows ε ε ε−1 ε H(t−1) from Theorem 3.1 that M (t) = L (t)(L1 M1)e . Consequently, ε −Ht H ε ε−1 ε M (t)e e = L (t)(L1 M1). (3.8)

Let U = U(M1, L1) = [U1|U0,U∞] satisfy (2.3) and (2.4) in which (M, L) is replaced by (M1, L1). From (2.8), we have

ε ε M1[U1,U0,U∞](I2ˆn ⊕ I` ⊕ εI`) = L1[U1,U0,U∞](Sb ⊕ εI` ⊕ I`). (3.9) 10 H −1 Plugging (3.9) into (3.8) and using the fact that e = U(Sb ⊕ I` ⊕ I`)U , we have

ε −Ht −1 ε −1 M (t)e U(I2ˆn ⊕ I` ⊕ εI`)U = L (t)U(I2ˆn ⊕ εI` ⊕ I`)U .

−Ht −Ht When ε approaches 0, it follows from (2.6) that M(t)e Π0 = L(t)Π∞. Since e commutes with Π0, we obtain assertion (3.6), and then (3.7). Corollary 3.5. Theorem 3.3 holds true if Eq. (3.1) is replaced by

H X˙ (t) = L(t)HJL(t) ,X(1) = X1.

Proof. It suﬃces to show that M(t)HJM(t)H = L(t)HJL(t)H . Using deﬁ- nitions of Π0 = Π0(M1, L1) and Π∞ = Π∞(M1, L1), we have M(t) = M(t)Π0, Ht L(t) = L(t)Π∞. It follows from (3.6) and the symplecticity of e that

H H H Ht Ht H H H M(t)HJM(t) = M(t)Π0HJ Π0 M(t) = L(t)Π∞e HJ (e ) Π∞L(t) H H H = L(t)Π∞HJ Π∞L(t) = L(t)HJL(t) .

Now, we study the invariance property (3.6). To this end, for given S1, S2 ∈

Sp(n), we let (M1, L1) ∈ SS1,S2 be regular with ind∞(M1, L1) 6 1. Let the idempotent matrices Π0 = Π0(M1, L1), Π∞ = Π∞(M1, L1) and H = H(M1, L1) be defined as in Definition 2.2. Consider the linear system with unknowns (M, L): ( MΠ = LΠ eHt, 0 ∞ (3.10) (M, L) ∈ SS1,S2 , where t ∈ R plays the role as a parameter. It is clear from Theorem 3.3 that the solution (M(t), L(t)) of the IVP (3.1) is contained in the manifold described by (3.10). In the following, we shall show that the consistency of Eq. (3.10) at specified t implies the uniqueness of the solution (M, L), for which the pair (M, L) is regular. From the

deﬁnition of SS1,S2 in (1.5), the second equation of (3.10) implies (M, L) = TS1,S2 (X) for some X = [Xij] ∈ H(2n). The linear system (3.10) can be rewritten as X12 0 IX11 Hbt S2U(I2ˆn ⊕ I` ⊕ 0) = S1U(e ⊕ 0 ⊕ I`). (3.11) X22 I 0 X21

The following lemma can be obtained by direct calculations. Lemma 3.6. Let

E11 = (I` ⊕ 0),E22 = (0 ⊕ I`), (3.12a) 1 2 V1 V1 V1 ≡ 1 = S1U, V2 ≡ 2 = S2U, (3.12b) V2 V2

j n×2n where Vi ∈ C for each 1 6 i, j 6 2. Then the linear system (3.11) is equivalent to the alternative form:

1 Hbt 1 Hbt X11 X12 −V2(e ⊕ E22) V1(e ⊕ E22) 2 = 2 . (3.13) X21 X22 V1(I2ˆn ⊕ E11) −V2(I2ˆn ⊕ E11)

11 Lemma 3.7. Let (A, B) be a regular pair with A, B ∈ Cn×n. Suppose that CA + DB = 0 and [C,D] ∈ Cn×2n is of full row rank. Then (D,C) is a regular pair. Proof. Since (A, B) is regular, there exists λ0 ∈ C such that A − λ0B is invertible and [A>,B>]> is of full column rank. Therefore, A I λ I I −λ I A 0 = [C,D] (A − λ B)−1 = [C,D] 0 0 (A − λ B)−1 B 0 0 I 0 I B 0 I = [C,D + λ0C] −1 . (3.14) B(A − λ0B)

It is easily seen that rank[C,D + λ0C] = rank[C,D] = n. Since the row vectors of −1 I [−B(A − λ0B) ,I] form a basis of left null space of −1 , it follows B(A − λ0B) from (3.14) that there is a nonsingular matrix W such that [C,D+λ0C] = W [−B(A− −1 λ0B) ,I]. Then D + λ0C is invertible and hence (D,C) is regular.

Theorem 3.8. Let (M1, L1) ∈ SS1,S2 be a regular pair with ind∞(M1, L1) 6 1 and U ≡ [U1,U0,U∞] = U(M1, L1). Suppose (M, L) is a solution of (3.10) at some t ∈ R. Then (i) (M, L) is regular; (ii) (M, L) is the unique solution of (3.10); (iii) It holds that

Hbt MU0 = 0, LU∞ = 0, MU1 = LU1e . (3.15)

Conversely, if (M, L) ∈ SS1,S2 satisﬁes (3.15), then (M, L) is a solution of (3.10). U(eHbt ⊕ E ) Proof. From (3.11), we have [−L, M] 22 = 0. Since rank([−L, M]) = U(I2ˆn ⊕ E11) Hbt 2n and (e ⊕ E22), (I2ˆn ⊕ E11) is regular, it follows from Lemma 3.7 that (M, L) is regular. Hence, assertion (i) holds. Next, we show that the linear system (3.10) has a unique solution. From Lemma 1 Hbt −V2(e ⊕ E22) 3.6, it suﬃces to show that the matrix 2 in (3.13) is invertible. V1(I2ˆn ⊕ E11) 1 Hbt 2n −V2(e ⊕ E22) Hbt Suppose that y ∈ C satisfying 2 y = 0. Let z1 = (e ⊕ E22)y V1(I2ˆn ⊕ E11) 1 2 and z2 = (I2ˆn ⊕E11)y. Then we have V2z1 = 0 and V1z2 = 0. Since the linear system 1 2 (3.13) is consistent, we obtain that V1z1 = 0 and V2z2 = 0. Hence, V1z1 = 0 and V2z2 = 0. From (3.12b), we have V1 and V2 are invertible, and hence, z1 = z2 = 0. Therefore, y = 0. Assertion (ii) follows. Since (3.11) is an alternative form of (3.10), and (3.11) is equivalent to (3.15)

whenever (M, L) ∈ SS1,S2 , assertion (iii) follows directly. Remark 3.9. Given two symplectic matrices S1 and S2, the linear system (3.10) may have no solution in SS1,S2 . We consider a simple example. Let S1 = S2 = I2, 0 π/2 0 1 H = and t = 1. Then eHt = . It is easily seen that (3.10) −π/2 0 −1 0 has no solution in SS1,S2 . From Theorem 3.8 (ii), the unique solution of (3.10) can be written as an implicit function (M(t), L(t)) for t on the set

TX = {t ∈ R | (3.10) has a solution at t}. (3.16) 12 Let X(t) = T −1 (M(t), L(t)), where (M(t), L(t)) is the solu- Remark 3.10. S1,S2 tion of (3.10) at t ∈ TX . We see that for each t ∈ TX , X(t) satisfies (3.13) and 1 Hbt −V2(e ⊕ E22) 2 is invertible, hence, X(t) is continuously differentiable. Conse- V1(I2ˆn ⊕ E11) quently, TX is open. Theorem 3.11. Suppose that (M(t), L(t)) is the solution of (3.10) for t ∈ (t˜ , t˜ ) ⊆ T , where t˜ < 1 < t˜ . Then X(t) = T −1 (M(t), L(t)) for t ∈ (t˜ , t˜ ) is 0 1 X 0 1 S1,S2 0 1 the solution of the IVP (3.1). Proof. Let Y (t) = T −1 (M(t), L(t)) for t ∈ (t˜ , t˜ ). From Remark 3.10, Y (t) is S1,S2 0 1 continuously differentiable. Suppose that X(t) for t ∈ (t0, t1) is the solution of the IVP (3.1), where (t0, t1) is the maximal interval. It follows from Theorem 3.3 that ˜ ˜ TS1,S2 (X(t)) is the solution of (3.10) at t ∈ (t0, t1). If (t0, t1) ⊆ (t0, t1), then the uniqueness of the solution of (3.10) implies that Y (t) = X(t) for t ∈ (t˜0, t˜1), and hence X(t) = T −1 (M(t), L(t)), for t ∈ (t˜ , t˜ ), is the solution of the IVP (3.1). Now S1,S2 0 1 ˜ ˜ ˜ we claim that (t0, t1) ⊆ (t0, t1). We prove the case t1 6 t1. On the contrary, suppose that t˜1 > t1. Then t1 ∈ (t˜0, t˜1) ⊆ TX , and hence, (M(t1), L(t1)) is the solution of (3.10) at t = t1. By the uniqueness of solution of (3.10), we have X(t) = Y (t) for t ∈ (t0, t1). We also note that Y˙ (t) is continuous at t1 ∈ (t˜0, t˜1). Therefore,

H H Y˙ (t1) − M(t1)HJM(t1) = lim Y˙ (t) − M(t)HJM(t) = 0. − t→t1

Hence, the solution X(t) of the IVP (3.1) can be extended to t1. This is a contradiction because (t0, t1) is the maximal interval of the IVP (3.1). Remark 3.12. Theorem 3.11 shows that the connected component of TX containing t = 1 coincides with the maximal interval of the IVP (3.1). Moreover, the phase portrait of the IVP (3.1) is the curve {(M(t), L(t)) deﬁned by (3.10) | t ∈ (t˜0, t˜1) ⊆ TX }. The solution of the IVP (3.1) can be extended to whole TX by using the so-called Grassmann manifold which will be studied in Subsection 3.3 for details.

3.2. Structure-preserving flow vs. Riccati differential equation. In this subsection, we shall show that the IVP (3.1) is governed by a Riccati differential equation (RDE). In addition, by adopting Radon’s Lemma (see Theorem 3.14), the structure-preserving flow can be represented as an explicit form. Throughout this subsection, we suppose that the assumptions in Theorem 3.3 are satisfied. First, −1 since S2 is symplectic and H is Hamiltonian, S2HS2 is also Hamiltonian, say

AS S HS−1 = , (3.17) 2 2 D −AH

n×n H H where A, S, D ∈ C with S = S and D = D. Suppose that X(t) = [Xij(t)]16i,j62, for t ∈ (t0, t1) and t0 < 1 < t1, is the solution of (3.1). We then have

˙ ˙ H H X11 X12 X12 0 −1 X12 X22 = S2HS2 J (3.18) X˙ 21 X˙ 22 X22 I 0 I H H −X12SX12 −X12SX22 + X12A = H H H H H H , −X22SX12 + A X12 −X22SX22 + X22A + A X22 + D 1 Xij(1) = Xij for 1 6 i, j 6 2. 13 That is, Xij(t) for 1 6 i, j 6 2 satisfy the coupled diﬀerential equations ˙ H X11 = −X12SX12, (3.19a) ˙ H X12 = −X12SX22 + X12A, (3.19b) ˙ H H H X21 = −X22SX12 + A X12, (3.19c) ˙ H H H X22 = −X22SX22 + X22A + A X22 + D, (3.19d)

1 with Xij(1) = Xij, where A, D and S are given in (3.17). Note that S, D and the 1 initial matrix X22 are Hermitian. From (3.19d), X22(t) is Hermitian for t ∈ (t0, t1). Therefore, by taking a time shift, W (t) = X22(t+1), t ∈ (t0 −1, t1 −1), is the solution of the RDE:

H W˙ (t) = −W (t)SW (t) + W (t)A + A W (t) + D,W (0) = W0, (3.20)

1 with W0 = X22. Remark 3.13. Suppose that W (t), for t ∈ (t0 − 1, t1 − 1) and t0 − 1 < 0 < t1 − 1, is a solution of the RDE (3.20). Using the fact X22(t) = W (t − 1), t ∈ (t0, t1), we can get X12(t) for t ∈ (t0, t1) by solving the linear diﬀerential equation (3.19b) 1 1 1H with X12(1) = X12. Since X21 = X12 , it follows from (3.19b) and (3.19c) that H X21(t) = X12(t) , for t ∈ (t0, t1). Finally, X11(t) for t ∈ (t0, t1) can be obtained directly from (3.19a). So, solving the IVP (3.1) is governed by solving the RDE (3.20). Theorem 3.14. [1, Radon’s Lemma] Let A, S, D ∈ Cn×n with SH = S and DH = D, then the following statements hold. (i) Let W (t) be a solution of the RDE (3.20) in the interval (t0 − 1, t1 − 1) containing zero. If Q(t) is a solution of the IVP Q˙ (t) = (SW (t) − A)Q(t), > > > with initial Q(0) = In and P (t) := W (t)Q(t), then Y (t) ≡ [Q(t) ,P (t) ] is the solution of the linear IVP

˙ > > Y (t) = HeY (t),Y (0) = [I,W0 ] , (3.21a) where −AS H = . (3.21b) e DAH

(ii) Let Y (t) ≡ [Q(t)>,P (t)>]> be the solution of (3.21). If Q(t) is invertible for −1 t ∈ (t0 − 1, t1 − 1) ⊂ R, then W (t) ≡ P (t)Q(t) is a solution of RDE (3.20). Remark 3.15. Using the deﬁnition of He in (3.21b), it follows from (3.17) that

−1 He = −(I ⊕ (−I))S2HS2 (I ⊕ (−I)). (3.22) −1 U1 U1 U1 U1 Therefore, if S2HS2 = Λ, then He = (−Λ). U2 U2 −U2 −U2 Corollary 3.16. Let Y (t) ≡ [Q(t)>,P (t)>]> and W (t) be the solution of (3.21) 1 and (3.20), respectively, with W0 = X22. If Q(t) is invertible, for t ∈ (t0 − 1, t1 − 1) and t0 − 1 < 0 < t1 − 1, then the solutions of (3.19d) and (3.19b) are

−1 1 −1 X22(t) = W (t − 1) = P (t − 1)Q(t − 1) and X12(t) = X12Q(t − 1) , (3.23)

H −H 1 respectively, for t ∈ (t0, t1). In addition, X21(t) = X12(t) = Q(t − 1) X21. 14 Proof. The detail of the proof can be found in [21]. −1 A? S? Let S1HS1 = H . From Corollary 3.5 and a similar calculation as D? −A? (3.18), we obtain that X11(t) and X21(t) satisfy

˙ H H H X11 = X11D?X11 + X11A? + A?X11 − S?, ˙ H H (3.24) X21 = X21D?X11 + X21A? ,

1 1 with X11(1) = X11 and X21(1) = X21. Similarly, by using the fact that the solution X11(t) is Hermitian and taking the time shift, t → t+1, we see that W?(t) = X11(t+1) is the solution of the RDE

˙ H 1 W?(t) = W?(t)D?W?(t) + W?(t)A? + A?W?(t) − S?,W?(0) = X11. (3.25)

> > > Let Y?(t) = [Q?(t) ,P?(t) ] be the solution of the linear diﬀerential equation

˙ 1> > Y?(t) = He?Y?(t),Y?(0) = [I,X11 ] , (3.26a) where

H −A? −D? −1 −1 He? ≡ = J S1HS1 J . (3.26b) −S? A?

? ? ? ? Suppose that Q?(t) is invertible for t ∈ (t0 − 1, t1 − 1) and t0 − 1 < 0 < t1 − 1. By Radon’s Lemma and Corollary 3.16, the solution X11(t), X21(t) of (3.24) can be formulated by

−1 1 −1 X11(t) = W?(t − 1) = P?(t − 1)Q?(t − 1) ,X21(t) = X21Q?(t − 1) , (3.27)

? ? respectively, for t ∈ (t0, t1). Comparing (3.22) and (3.26b) yields that He? and −He are similar. The nonsingularity of Q(t) and Q?(t) plays an important role to determine whether X22(t) and X11(t) exist, respectively. The following theorem claims that both Q(t) and Q?(t) are invertible simultaneously. Theorem 3.17. Let Q(t), P (t), Q?(t) and P?(t) be the matrix functions given in (3.21) and (3.26), respectively. Then we have

{t ∈ R| det(Q(t)) 6= 0} = {t ∈ R| det(Q?(t)) 6= 0}. (3.28)

ˆ ˆ −1 1 −1 In addition, if t ∈ R such that det(Q(t)) = 0, then kP (t)Q(t) k, kX12Q(t) k, −1 1 −1 ˆ kP?(t)Q?(t) k, and kX21Q?(t) k → ∞ as t → t. Proof. Let Π0,Π∞, U ≡ [U1|U0,U∞] and H be given in Definition 2.2 that satisfy Ht (3.5). Using the facts that S1, S2 and e are symplectic and applying (3.21), (3.22) and (3.26), we have Q(t) −Ht −1 I 0 I Q(t) = [I, 0] = [I, 0]S2e S2 1 P (t) 0 −I X22 X1 X1 = [I, 0]S e−HtS−1J 22 = [I, 0]S J (eHt)H SH 22 , (3.29) 2 2 I 2 2 I Q?(t) Ht H I Q?(t) = [I, 0] = −[0,I]S1e JS1 1 . (3.30) P?(t) X11 15 ˆ ˆ −1 Suppose that t ∈ R such that det(Q(t)) = 0. We first claim that limt→tˆ kP (t)Q(t) k = 1 −1 ˆ > ˆ > > Hetˆ 1 > limt→tˆ kX12Q(t) k = ∞. Since [Q(t) ,P (t) ] = e [I,X22] is of full column rank ˆ −1 and Q(t) is singular, it is easily seen that limt→tˆ kP (t)Q(t) k = ∞. Now, we show 1 −1 ˆ that limt→tˆ kX12Q(t) k = ∞. Since Q(t) is continuous and Q(t) is singular, it 1 ˆ suffices to show that X12x0 6= 0, where Q(t)x0 = 0 with x0 6= 0. We prove it by 1 contradiction. Suppose that X12x0 = 0. Since (M1, L1) ∈ SS1,S2 , Eq. (3.5) can be written in the form 1 1 X12 0 IX11 H 1 S2Π0 = 1 S1Π∞e . (3.31) X22 I 0 X21

1 1H 1 Using the facts that X12 = X21 and X12x0 = 0, it follows from the second row of H 1 H 1 (3.31) that x0 [X22,I]S2Π0 = 0. Since Π0[U1,U0] = [U1,U0], we have x0 [X22,I]S2[U1,U0] = 0. Using the deﬁnition of H in (2.5) yields

Hbt H 1 Ht H 1 e 0 −1 x0 [X22,I]S2e = x0 [X22,I]S2[U1|U0,U∞] [U1|U0,U∞] 0 I2` H 1 −1 = 0, 0, x0 [X22,I]S2U∞ [U1|U0,U∞] ,

H H 1 Ht which is independent of the parameter t. Therefore, we may denote z0 = x0 [X22,I]S2e . Post-multiplying x0 to (3.29), it holds that 1 H X Q(t)x = [I, 0]S J eHt SH 22 x = [I, 0]S J z 0 2 2 I 0 2 0 which is independent of the parameter t. Because Q(0) = I and x0 6= 0, we have Q(tˆ)x0 = Q(0)x0 6= 0. This contradicts that Q(tˆ)x0 = 0. −1 1 −1 Now, we show that limt→tˆ kP?(t)Q?(t) k = limt→tˆ kX21Q?(t) k = ∞. Us- 1 −1 H 1 −1 H ing the fact that X21Q?(t) = X21(t + 1) = X12(t + 1) = (X12Q(t) ) , we have 1 −1 ˆ −1 limt→tˆ kX21Q?(t) k = ∞. Consequently, Q?(t) is singular. Then limt→tˆ kP?(t)Q?(t) k = −1 ∞ can be proven by the similar argument for limt→tˆ kP (t)Q(t) k = ∞. This proves the inclusion {t ∈ R| det(Q(t)) = 0} ⊆ {t ∈ R| det(Q?(t)) = 0}. The conclusion can be shown accordingly by (3.30). Hence, (3.28) holds true. Now, let

TW = {t ∈ R| Q(t) is invertible}. (3.32)

Theorem 3.17 enables us to write the set TW in an alternative form TW = {t ∈ R| Q?(t) is invertible}. From (3.29), det(Q(t)) is a nonzero analytic function, and hence, the zeros of det(Q(t)) are isolated. It follows TW is the set that R subtracts some isolated points, and hence, TW is a union of open intervals, say [ TW = (tˆk, tˆk+1). (3.33) k∈Z

Here det(Q(tˆk)) = 0 for each k and ··· < tˆ−1 < tˆ0 < tˆ1 < ··· . Since Q(0) = I, it implies that 0 ∈ TW . For convenience, say 0 ∈ (tˆ0, tˆ1). Therefore, from Radon’s Lemma it follows that (tˆ0, tˆ1) is the maximal interval of the RDEs (3.20) and (3.25). In Subsection 3.3, we shall extend the domain of W (t) and W?(t) to whole TW . 3.3. The extension of structure-preserving ﬂow: the phase portrait on Grassmann manifolds. Let Gn(C2n) be the Grassmann manifold that consists of n-dimensional subspaces of a 2n-dimensional space, equipped with an appropriate topology (see e.g., [1]). Intrinsically, Gn(C2n) can be written as 16 A A Gn( 2n) = Im | A, B ∈ n×n and rank = n . C B C B Here Im[A>,B>]> denotes the column space spanned by [A>,B>]>. It is easily seen n×n n 2n > > n 2n that C can be embedded into G (C ) by ψ(W ) = Im [I,W ] . Let G0 (C ) = > > > n 2n n×n n 2n n×n Im [A ,B ] ∈ G (C )| A ∈ C is invertible . Then G0 (C ) = ψ(C ) is the image of ψ. Note that the Grassmann manifold Gn(C2n) is a compact analytic 2 n 2n n 2n manifold of dimension n and that G0 (C ) is an open dense subset of G (C ). Radon’s Lemma leads us to consider a natural extension of the solution of the RDE (3.20) in Cn×n to a function on the Grassmann manifold Gn(C2n), via the process by the embedding ψ(W (t)) = Im([I,W (t)>]>) = Im([Q(t)>,P (t)>]>). Hence, the solution of the RDE (3.20) on Gn(C2n) is just the solution of (3.21). Note that the maximal interval of the solution of (3.21) is R. In addition, the representation of Theorem 3.14 (ii) holds not only for all t ∈ (tˆ0, tˆ1) but also for t ∈ TW deﬁned in (3.32). Hence, the extended solution of the RDE (3.20) is

−1 W (t) = P (t)Q(t) , for t ∈ TW

> > > n 2n where [Q(t) ,P (t) ] is the solution of (3.21). Here, ψ(W (t)) ∈ G0 (C ) for t ∈ TW . In the case t 6∈ TW , i.e., t = tˆk for some k ∈ Z, W (t) does not exist but > > > n 2n n 2n > > > Het > > Im [Q(t) ,P (t) ] ∈ G (C ) \ G0 (C ). Since [Q(t) ,P (t) ] = e [I,W0 ] , both Q(t) and P (t) are analytic functions of t, and hence, W (t) is meromorphic. We note that the unboundedness of TW implies that the limit, limt→∞ W (t), is mean- ingful. The asymptotic phenomena of the phase portrait of the RDE (3.20) can be investigated by using the extended solution of the RDE. Theorem 3.17 shows that Q(t) and Q?(t) are simultaneously invertible, where > > > > > > Y (t) = [Q(t) ,P (t) ] and Y?(t) = [Q?(t) ,P?(t) ] are the solutions of (3.21) and (3.26), respectively. From Corollary 3.16 and (3.27), the extended solution, X(t) =

[Xij(t)]16i,j62, of the IVP (3.1) can be deﬁned as −1 1 −1 X11(t) = P?(t − 1)Q?(t − 1) ,X12(t) = X12Q(t − 1) , 1 −1 −1 (3.34) X21(t) = X21Q?(t − 1) ,X22(t) = P (t − 1)Q(t − 1) , for t ∈ TW + 1, where TW + 1 denotes the set

TW + 1 ≡ {t + 1|t ∈ TW } = {t ∈ R| Q(t − 1) is invertible}. (3.35) In Remark 3.12, we demonstrate that the maximal interval of the IVP (3.1), i.e., the maximal interval of TW + 1 containing 1, coincides with the connected component of TX containing 1. In the following theorem we will show that TW + 1 = TX and

(M(t), L(t)) = TS1,S2 (X(t)) satisﬁes (3.10) for t ∈ TW +1, where X(t) is the extended solution of the IVP (3.1), and vice versa. Theorem 3.18. Suppose the assumptions of Theorem 3.3 hold. (i) If X(t), for t ∈ TW + 1, is the extended solution of the IVP (3.1), then

(M(t), L(t)) = TS1,S2 (X(t)) satisﬁes (3.10) for t ∈ TW + 1; (ii) TW + 1 = TX where TX is deﬁned in (3.16); (iii) if (M(t), L(t)) is the solution of (3.10) for t ∈ T , then X(t) = T −1 (M(t), L(t)) X S1,S2 is the extended solution of the IVP (3.1).

Proof. We ﬁrst prove assertion (i). Suppose that X(t) = [Xij(t)]16i,j62 for t ∈ TW + 1, deﬁned in (3.34), is the extended solution of the IVP (3.1). Since X22(t) H is Hermitian and X21(t) = X12(t) , it holds that

−H 1H H [X21(t),X22(t)] = Q(t − 1) [X12 ,P (t − 1) ], (3.36) 17 where [Q(t)>,P (t)>]> is the solution of the IVP (3.21). Using the deﬁnitions of H and He in (2.5) and (3.22), respectively, we have Q(t − 1) I 0 −Hb(t−1) −1 −1 I = S2U(e ⊕ I2`)U S2 1 . (3.37) P (t − 1) 0 −I −X22

−H(t−1) −Hb(t−1) −1 Since S2 and e = U(e ⊕ I2`)U are symplectic, we have

−Hb(t−1) −1 −1 −H −H Hb(t−1) H H H JS2U(e ⊕ I2`)U S2 = S2 U (e ⊕ I2`) U S2 J . Applying the last equation to (3.37) it follows that H H P (t − 1) Hb(t−1) H H H I U S2 = −(e ⊕ I2`) U S2 J 1 Q(t − 1) −X22 X1 = (eHb(t−1) ⊕ I )H UH SH 22 . (3.38) 2` 2 I

Using the fact that (M1, L1) ∈ SS1,S2 satisfies (3.5), we have 1 1 X12 0 IX11 Hb 1 S2U(I2ˆn ⊕ E11) = 1 S1U(e ⊕ E22), (3.39) X22 I 0 X21 1H 1 where E11 and E22 are defined in (3.12a) andn ˆ = n − `. Since X22 = X22 and 1H 1 X21 = X12, it follows from (3.38) and (3.39) that H H Q(t − 1) P (t − 1) (I ⊕ E )[V2 , V2 ] = (I ⊕ E )UH SH 2ˆn 11 2 1 P (t − 1) 2ˆn 11 2 Q(t − 1) X1 = (eHb(t−1) ⊕ I )H (I ⊕ E )H UH SH 22 2` 2ˆn 11 2 I Hb(t−1) H Hb H H H 0 Hbt H 1H 1 = (e ⊕ I2`) (e ⊕ E22) U S1 1H = (e ⊕ E22) V2 X12, X21 where V1 and V2 are defined in (3.12b). We then have 1 H H X H [−(eHbt ⊕ E )H V1 , (I ⊕ E )V2 ] 12 = −(I ⊕ E )V2 Q(t − 1). 22 2 2ˆn 11 1 P (t − 1) 2ˆn 11 2 1 Hbt −V2(e ⊕ E22) Combining the last equation and (3.36), we obtain [X21(t),X22(t)] 2 = V1(I2ˆn ⊕ E11) 2 −V2(I2ˆn ⊕E11). Therefore, the equality of the second row of (3.13) holds. The equality of the first row can be accordingly obtained by using the formulas for X11(t) and H > > > X12(t) = X21(t) in (3.34) and the solution Y?(t) = [Q?(t) ,P?(t) ] of the linear differential equation (3.26). Since (3.13) is equivalent to (3.10) by Lemma 3.6, this shows assertion (i). Now we prove assertion (ii). From (i), we have TW + 1 ⊆ TX . From (3.33) and S (3.35), we obtain that TW + 1 = (tˆk + 1, tˆk+1 + 1) ⊆ TX . For each k ∈ , we k∈Z Z have that (tˆk + 1, tˆk+1 + 1) ⊆ TX . Choosing a point tk+1/2 ∈ (tˆk + 1, tˆk+1 + 1), it follows from (i) that (M(tk+1/2), L(tk+1/2)) = TS1,S2 (X(tk+1/2)) is the solution of (3.10) at t = tk+1/2. A similar argument to Theorem 3.11 and Remark 3.12 shows that (tˆk + 1, tˆk+1 + 1) is the connected component of TX containing tk+1/2. Hence, TW + 1 = TX . Now we prove assertion (iii). From Theorem 3.8 it follows that the solution (M(t), L(t)) of (3.10) is unique for each t ∈ TX . Therefore, assertions (i) and (ii) lead to the fact that X(t) = T −1 (M(t), L(t)) is the extended solution. This completes S1,S2 the proof. 18 3.4. Structure-preserving flow vs. SDA. The structure-preserving flow

(M(t), L(t)) = TS1,S2 (X(t)) ∈ SS1,S2 with the initial (M(1), L(1)) = (M1, L1) has been constructed in Theorem 3.3, where X(t) for t ∈ TX is the extended solution of the IVP (3.1). In addition, Theorem 3.18 shows the phase portrait of this ﬂow is actually the curve CM1,L1 = {(M(t), L(t)) | (M(t), L(t)) is a solution of (3.10) at t ∈ TX }.

On the other hand, CM1,L1 is actually the so-called Structuring-Preserving Curve

constructed in the work [22] for considering the case S2, in which CM1,L1 passes through the iterations of the SDA-2. We now show the Main Result II which is a generalization of the work [22]. ∞ Theorem 3.19 (Main Result II). Let {(Mk, Lk)}k=1 be the sequence generated by SDA-1 or SDA-2 with (M1, L1) ∈ S1 or S2, respectively, and ind∞(M1, L1) 6 1. k−1 k−1 Then (Mk, Lk) = (M(2 ), L(2 )), where (M(t), L(t)) = TS1,S2 (X(t)) and X(t) is the extended solution of the IVP (3.1). Proof. By the Eigenvector-Preserving Property for SDAs, we have

Hb2k−1 MkU0 = 0, LkU∞ = 0 and MkU1 = LkU1e (3.40)

Hb for each k ∈ N, where the initial pair (M1, L1) satisﬁes (2.4) with Sb = e . By k−1 k−1 applying Theorem 3.18 (iii) to (3.40), we have (Mk, Lk) = (M(2 ), L(2 )), and hence, the assertion follows. 4. Application to the convergence analysis of SDA. The structure-preserving ﬂows are governed by the RDEs of the compact form:

˙ > > W (t) = [−W (t),I]H [I,W (t) ] ,W (0) = W0, (4.1)

where H is a Hamiltonian matrix. By Theorem 3.14, the extended solution W (t; H ,W0) = −1 P (t; H ,W0)Q(t; H ,W0) where Q(t; H ,W0) and P (t; H ,W0) are of the form Q(t; H ,W0) H t I Y (t; H ,W0) ≡ = e . (4.2) P (t; H ,W0) W0

Speciﬁcally, RDEs (3.20) and (3.25) are of the compact form as in (4.1) with H = −1 −1 −1 He = −(In ⊕ −In)S2HS2 (In ⊕ −In) and H = He? = J S1HS1 J given in (3.22) −1 and (3.26b), respectively. Applying the relation between He and S2HS2 together with (4.2), Eqs. (3.23) and (3.27) have the alternative form:

1 −1 1 X22(t) = W (t − 1; He,X22) = −W (t − 1; −S2HS2 , −X22), −1 1 −1 1 −1 = −P (t − 1; −S2HS2 , −X22)Q(t − 1; −S2HS2 , −X22) , −1 1 −1 1 −1 = −P (−t + 1; S2HS2 , −X22)Q(−t + 1; S2HS2 , −X22) , (4.3a) −1 1 = −W (−t + 1; S2HS2 , −X22) 1 −1 1 −1 X12(t) = X12Q(−t + 1; S2HS2 , −X22) , 1 1 1 −1 X11(t) = W (t − 1; He?,X11) = P (t − 1; He?,X11)Q(t − 1; He?,X11) , 1 1 −1 (4.3b) X21(t) = X21Q(t − 1; He?,X11) ,

for t ∈ TW + 1. We conclude that (i) the large time behaviors of X22(t), X12(t) as t → ∞ are determined by −1 1 −1 1 −1 W (t; S2HS2 , −X22) and Q(t; S2HS2 , −X22) as t → −∞; (ii) the large time behaviors of X11(t), X21(t) as t → ∞ are determined by 1 1 −1 W (t; He?,X11) and Q(t; He?,X11) as t → ∞. 19 −1 Note that Hamiltonian matrices S2HS2 and He? are symplectically similar. By assertions (i) and (ii) above, we see that the asymptotic behaviors of X22(t), X12(t) and X11(t), X21(t) as t → ∞ are governed by −1 1 Ht −1 I Y (t; S2HS2 , −X22) = S2e S2 1 (as t → −∞), (4.4a) −X22 and 1 −1 Ht −1 I Y (t; He?,X11) = J S1e S1 J 1 (as t → ∞), (4.4b) X11 respectively. For both cases in (4.4a) and (4.4b), eHt is involved. Therefore, for a given Hamiltonian matrix H , we are interested in the study of the asymptotic behavior of the solution, W (t) = P (t)Q(t)−1, of the RDE (4.1) and Q(t)−1 as t → ±∞. Due to the special structure of the Hamiltonian matrix H , rather than applying a Jordan canonical form to H , we shall adopt the Hamiltonian Jordan canonical form for studying the asymptotic behavior of RDEs. A canonical form of a Hamiltonian matrix under symplectic similarity transformations has been investigated in [26]. Let Nk be the k × k nilpotent matrix, and let Nk(λ) = λIk + Nk be the Jordan block of the eigenvalue λ with size k. In the following, we shall apply the asymptotic analysis of (4.3). Throughout this section, we ﬁx (S1, S2) = (I,I) (the S1 class) or (J , −I) (the S2 class) and let the pair (M1, L1) ∈ S1 or S2 be regular with ind∞(M1, L1) 6 1. 4.1. Proof of Main Result III (i). To simplify the proof, we suppose that H has no other eigenvalues than λ and −λ¯ and there is only one Jordan block for each of the two eigenvalues. Denote r = Re(λ) > 0. Then from [26], there is a −1 H symplectic matrix S such that J := S HS = Nn(λ) ⊕ (−Nn(λ) ). Let S− = S2S, ± ± −1 U1 V1 ± ± ± ± n×n S+ = J S1S. Partition S± = ± ± , where U1 ,U2 ,V1 ,V2 ∈ C . Let U2 V2

− + W1 −1 I W1 −1 I 2n×n − = S− 1 , + = S+ 1 ∈ C . (4.5) W2 −X22 W2 X11 Then we have following results. + − + − Lemma 4.1. If U1 , V1 , W1 and W2 are invertible, then

− − −1 −2rt 2(n−1) −rt n−1 X22(t) = −V2 (V1 ) + O(e t ),X12(t) = O(e t ), (4.6a) + + −1 −2rt 2(n−1) −rt n−1 X11(t) = U2 (U1 ) + O(e t ),X21(t) = O(e t ), (4.6b) as t → ∞. Proof. We only prove (4.6a). Eq. (4.6b) can be obtained similarly. Since J = H Jt λt −λt¯ −H Nnt Nn(λ) ⊕ (−Nn(λ) ), we have e = (e Φn) ⊕ (e Φn ), where Φn ≡ Φn(t) = e . From (4.4a), we have

− − λt − Q(t) −1 1 U1 V1 e ΦnW1 ≡ Y (t; S2HS2 , −X22) = − − −λt¯ −H − . (4.7) P (t) U2 V2 e Φn W2

−1 −Nnt It is well known that Φn (t) = e is a polynomial of order n − 1 and hence, −H n−1 λ¯(t−1) − −H − −2r(t−1) − − kΦn k = O(t ). Since Q(−t + 1) = e (V1 Φn W2 + e U1 ΦnW1 ), if − − 1 −1 −rt n−1 V1 and W2 are invertible, then we have X12(t) = X12Q(−t + 1) = O(e t ) 20 as t → ∞. From (4.3a), the second assertion in (4.6a) follows. Using the facts that − − Φn, V1 and W2 are invertible, we obtain from (4.3a) and (4.7) that

−1 X22(t) = −P (−t + 1)Q(−t + 1) − −2r(t−1) − − − −1 H − −2r(t−1) − − − −1 H −1 = −(V2 + e U2 ΦnW1 (W2 ) Φn )(V1 + e U1 ΦnW1 (W2 ) Φn ) − − −1 −2rt 2(n−1) = −V2 (V1 ) + O(e t ),

as t → ∞. We complete the proof. Applying the results of Lemma 4.1 to Theorem 3.19 directly, the following theorem follows. ∞ Theorem 4.2. Let {(Mk, Lk)}k=1 be the sequence generated by the SDA-1 or ¯ + SDA-2. Suppose that H has only eigenvalues λ and −λ with r = Re(λ) > 0. If U1 , − + − V1 , W1 and W2 are invertible, then

k − − −1 −r2k 2nk k −r2k−1 nk X22 = −V2 (V1 ) + O(e 2 ),X12 = O(e 2 ), k + + −1 −r2k 2nk k −r2k−1 nk X11 = U2 (U1 ) + O(e 2 ),X21 = O(e 2 ),

as k → ∞, where [Xk ] ≡ T −1 (M , L ). ij 1≤i,j≤2 S1,S2 k k This theorem shows that the SDA exhibits a quadratic convergence whenever none of nonzero eigenvalues of H are pure imaginary. A similar convergence analysis has been carried out in [3, 16].

4.2. Proof of Main Result III (ii). We only consider that H ∈ C2n×2n has only one pure imaginary eigenvalue iα having one Jordan block of size 2n. From [26], H −1 Nn(iα) βenen there is a symplectic matrix S such that J := S HS = H , 0 −(Nn(iα)) −1 2n×2n where β ∈ {−1, 1}. Note that N2n(iα) = ΘJΘ , where Θ = In ⊕ (−βPn) ∈ R j and Pn(j, n + 1 − j) = (−1) , j = 1, ··· , n. We denote

 tn t(n+1) t2n−1  n! (n+1)! ··· (2n−1)!  (n−1) n . (2n−2)   t t .. t   (n−1)! n! (2n−2)!  Γn ≡ Γn(t) =   , (4.8)  ......   . . . .  t1 tn 1! ······ n!

Γbn ≡ Γbn(t) = ΓnPn, Jt iαt −1 N2nt Jt iαt Φn(t) −βΓbn(t) Then e = e Θ e Θ has the structured form e = e −H , 0 (Φn(t)) Nnt where t ∈ R,Φn(t) = e and Γbn(t) is deﬁned in (4.8). ± ± −1 U1 V1 ± ± Let S− = S2S, S+ = J S1S. Partition S± = ± ± , where Uj ,Vj ∈ U2 V2 n×n ± ± C , j = 1, 2. Let W1 and W2 be deﬁned as (4.5). Then we have following results. ± ± Lemma 4.3. If U1 , W2 are invertible, then

− − −1 −1 −1 X22(t) = −U2 (U1 ) + O(t ),X12(t) = O(t ), (4.9a) + + −1 −1 −1 X11(t) = U2 (U1 ) + O(t ),X21(t) = O(t ), (4.9b)

as t → ∞. 21 Proof. We only prove (4.9a). Eq. (4.9b) can be obtained similarly. From the structure form of eJt above and (4.4a) we have

− − − 2n−1 − Q(t) −1 1 iαt U1 V1 ΦnW1 − βΓbn W2 ≡ Y (t; S2HS2 , −X22) = e − − −H − . P (t) U2 V2 Φn W2

− − − −1 2n−1 −1 Since W2 is invertible, we let Ω(t) = (ΦnW1 (W2 ) − βΓbn ) . Using the fact −H −1 −1 that Φn Ω(t) = O(t ) as t → −∞ (the detail can be found [21]), we have

− − − −1 Q(t) − −1 iαt U1 V1 I iαt U1 + O(t ) (W2 ) Ω(t) = e − − −1 = e − −1 , P (t) U2 V2 O(t ) U2 + O(t )

−1 iα(t−1) − −1 − −1 −1 −1 as t → −∞. Then Q(−t+1) = e (W2 ) Ω(−t+1)(U1 +O(t )) = O(t ) −1 − because Ω(−t + 1) = O(t ) as t → ∞. Since U1 is invertible, from (4.3a) we have −1 − − −1 −1 1 X22(t) = −P (−t + 1)Q(−t + 1) = −U2 (U1 ) + O(t ) and X12(t) = X12Q(−t + 1)−1 = O(t−1), as t → ∞. We complete the proof. Applying the results of Lemma 4.3 to Theorem 3.19, the Theorem 4.4 follows. ∞ Theorem 4.4. Let {(Mk, Lk)}k=1 be the sequence generated by the SDA-1 or SDA-2. Suppose that H has only one pure imaginary eigenvalue iα with partial mul- ± ± tiplicity 2n. If U1 , W2 are invertible, then

k − − −1 −k k −k X22 = −U2 (U1 ) + O(2 ),X12 = O(2 ), k + + −1 −k k −k X11 = U2 (U1 ) + O(2 ),X21 = O(2 ), as k → ∞, where [Xk ] ≡ T −1 (M , L ). ij 1≤i,j≤2 S1,S2 k k This theorem shows that the SDA exhibits a linear convergence whenever the partial multiplicities corresponding to nonzero pure imaginary eigenvalues of H are even. A similar convergence analysis has been proven in [20]. 4.3. Proof of Main Result III (iii). Suppose that H has only one pure imaginary eigenvalue iα with two odd partial multiplicities 2n1 + 1 and 2n2 + 1. Note that n1 + n2 + 1 = n. By [26], there is a symplectic matrix S such that R D J := S−1HS = , where 0 −RH  √  N (iα) 0 − 2 e √  0 0 e  n1 √2 n1 2 n1  2  R = 0 N (iα) − e ,D = iβ  0 0 −en2  , (4.10)  n2 2 n2  2 −eH eH 0 0 0 iα n1 n2

± ± ± ± −1 U1 u1 V1 v1 β ∈ {−1, 1}. Let S− = S2S, S+ = J S1S. Partition S± = ± ± ± ± , U2 u2 V2 v2 ± ± n×(n−1) ± ± n ± ± where Uj ,Vj ∈ C and uj , vj ∈ C for j = 1, 2. Let W1 and W2 be deﬁned as (4.5). The following lemma is useful for the reduction of Y (t). The detail of proof can be found in [21]. ± ± ± ± ± ± −1 W1,1 w1,2 Lemma 4.5. Let W2 be invertible and W := W1 (W2 ) = ± ± , w2,1 w2,2 ± (n1+n2)×(n1+n2) ± ± H n1+n2 ± where W1,1 ∈ C , w1,2, (w2,1) ∈ C and w2,2 ∈ C. Let

± ± ± ± ± ± U1 U1 fu u1 + fv v1 U± ≡ ± = ± ± ± ± ± , (4.11) U2 U2 fu u2 + fv v2 22 ± ± n1 ± ± ± n1 ± where (fu , fv ) = (−1) (w2,2, 1) if n is odd; otherwise (fu , fv ) = (−1) iβ(−1, w2,2). Then there exist nonsingular matrices Ω±(t) of the form

O(t−2) O(t−1) Ω (t) = , as t → ±∞, (4.12) ± 0 e−iαt satisfying

−1 1 − −1 −1 Y (t; S2HS2 , −X22)(W2 ) Ω−(t) = U− + O(t ), as t → −∞ 1 + −1 −1 (4.13) Y (t; He?,X11)(W2 ) Ω+(t) = U+ + O(t ), as t → ∞.

Then we have following results. ± ± Lemma 4.6. If U1 , W2 are invertible, then

− − −1 −1 X22(t) = −U2 (U1 ) + O(t ), iα(t−1) 1 − −1 H − −1 −1 (4.14a) X12(t) = e X12(W2 ) enen (U1 ) + O(t ), + + −1 −1 X11(t) = U2 (U1 ) + O(t ), −iα(t−1) 1 + −1 H + −1 −1 (4.14b) X21(t) = e X21(W2 ) enen (U1 ) + O(t ),

± as t → ∞, where Uj for j = 1, 2 are deﬁned in (4.11). Proof. We only prove (4.14a). Eq. (4.14b) can be obtained similarly. Denote > > > −1 1 [Q(t) ,P (t) ] = Y (t; S2HS2 , −X22). From Lemma 4.5 and (4.3a), (4.4a) we have −1 − − −1 −1 X22(t) = −P (−t + 1)Q(−t + 1) = −U2 (U1 ) + O(t ), as t → ∞. Using the iα(t−1) H −1 structure of Ω−(t) in (4.12), we see Ω−(−t + 1) = e enen + O(t ) as t → ∞. From (4.13) it follows that

−1 − −1 − −1 −1 Q(−t + 1) = (W2 ) Ω−(−t + 1)(U1 + O(t )) iα(t−1) − −1 H − −1 −1 = e (W2 ) enen (U1 ) + O(t ),

1 −1 iα(t−1) 1 − −1 H − −1 as t → ∞. Hence, X12(t) = X12Q(−t + 1) = e X12(W2 ) enen (U1 ) + O(t−1) as t → ∞. Applying Lemma 4.6 to Theorem 3.19, the Theorem 4.7 follows. ∞ Theorem 4.7. Let {(Mk, Lk)}k=1 be the sequence generated by the SDA-1 or SDA-2. Suppose that H has only one eigenvalue iα with two odd partial multiplicities ± ± 2n1 + 1 and 2n2 + 1. If U1 , W2 are invertible, then

k − − −1 −k k iα(2k−1−1) 1 − −1 H − −1 −k X22 = −U2 (U1 ) + O(2 ),X12 = e X12(W2 ) enen (U1 ) + O(2 ), k + + −1 −k k −iα(2k−1−1) 1 + −1 H + −1 −k X11 = U2 (U1 ) + O(2 ),X21 = e X21(W2 ) enen (U1 ) + O(2 ), as k → ∞, where [Xk ] ≡ T −1 (M , L ). ij 1≤i,j≤2 S1,S2 k k 5. Numerical experiments. In this section, we show some numerical experiments to demonstrate the eigenvalue eﬀects in Theorems 4.2 and 4.7. Here we ﬁx (S1, S2) = (I,I) (the S1 class). Example 5.1. Consider the Hamiltonian Jordan canonical form J = N2(λ) ⊕ H (−N2(λ) ), where λ = r + iα and r > 0. We construct the Hamiltonian matrix H = SJS−1 ∈ R4×4, where S is a randomly generated symplectic matrix. Using the H 1 1 1 1 2×2 symplectic matrix e , we can construct initial matrices X11,X12,X21,X22 ∈ R 1 1 > 1 1 H with X12 = (X21) and X11,X22 being symmetric such that M1 = L1e , where 1 (M1, L1) = TS1,S2 ([Xij]). Because the S1 class is considered, S− = S. Then X12(t) 23 3 1.4

2.5 1.2 1 - 1 ) || - 1 2 ( V - 2 0.8

1.5 (t) || 12 0.6 || X (t)+V 22 1 || X 0.4

0.5 0.2

0 0 0 1 2 3 4 5 10 10 10 10 10 10 0 1 2 3 4 5 t 10 10 10 t 10 10 10

− − −1 4 Fig. 5.1. kX22(t) + V2 (V1 ) k and kX12(t)k are plotted for 1 6 t 6 1.5 × 10 with r = 0.1 (blue line), 0.01 (red line) and 0.001 (green line).

and X22(t) can be computed by the formulas in (4.3a). Here, we let α = 0.801 be − − −1 ﬁxed, and vary r = 0.1, 0.01, and 0.001. We then plot kX22(t) + V2 (V1 ) k and 4 kX12(t)k for 1 6 t 6 1.5 × 10 in Figure 5.1. The blue, red and green lines in Figure 5.1 represents for the residuals with respect to r = 0.1, 0.01, and 0.001, respectively. k − − −1 k The circles on the lines represent for the residuals kX22 + V2 (V1 ) k and kX12k for the SDA at k = 3, 6, 9, 12 and 15. In each case, we can see that the convergence occurs, but the time for the residual bounded by the tolerance is postponed when r is smaller. R D Example 5.2. Consider the Hamiltonian Jordan canonical form J = , 0 −RH where R and D have form in (4.10) with α = 1.771, n1 = 2, n2 = 1, n = n1 +n2 +1 = 4 and β = 1. We construct the Hamiltonian matrix H = SJS−1 ∈ R8×8, where S is a randomly generated symplectic matrix. Using the symplectic matrix eH, we can con- 1 1 1 1 4×4 1 1 > 1 1 struct initial matrices X11,X12,X21,X22 ∈ R with X12 = (X21) and X11,X22 H 1 being symmetric such that M1 = L1e , where (M1, L1) = TS1,S2 ([Xij]). Therefore, S− = S. Then X12(t) and X22(t) can be computed by the formulas in (4.3a). We − − −1 iα(t−1) 1 − −1 H − −1 then plot kX22(t) + U2 (U1 ) k and kX12(t) − e X12(W2 ) enen (U1 ) k for 3 1 6 t 6 10 in Figure 5.2 (a). We can see that both of them approach 0 as t → ∞. It is shown in Lemma 4.6 that X12(t) approaches a rank-one periodic function with period 2π/α. In Figure 5.2 (b), we plot the phase portrait of the (1, 1)-entry of X12(t) that illustrates how it approaches a limit cycle. Acknowledgment. We thank the Editor-in-Chief, Professor Dianne P. O’ Leary, and the two anonymous referees for their careful reading, valuable comments and suggestions on the manuscript.

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